TITLE: Problems that become easier in a more general form
QUESTION [96 upvotes]: When solving a problem, we often look at some special cases first, then try to work our way up to the general case.
It would be interesting to see some counterexamples to this mental process, i.e. problems that become easier when you formulate them in a more general (or ambitious) form.

Motivation/Example
Recently someone asked for the solution of $a,b,c$ such that $\frac{a}{b+c} = \frac{b}{a+c} = \frac{c}{a+b} (=t).$
Someone suggested writing this down as a system of linear equations in terms of $t$ and solving for $a,b,c$. It turns out that either (i) $a=b=c$ or (ii) $a+b+c=0$.
Solution (i) is obvious from looking at the problem, but (ii) was not apparent to me until I solved the system of equations.
Then I wondered how this would generalize to more variables, and wrote the problem as:
$$
\frac{x_i}{\sum x - x_i} = \frac{x_j}{\sum x - x_j} \quad \forall i,j\in1,2,\dots,n
$$
Looking at this formulation, both solutions became immediately evident without the need for linear algebra (for (ii), set $\sum x=0$ so that each denominator cancels out with its numerator).

REPLY [4 votes]: Generalization comes up a lot when doing induction.  For example, 
$$\forall n ~~ \sum_{k=0}^n 2^{-k} \le 2$$
is difficult to prove directly using induction on $n$.  However, if you generalize to a stronger statement:
$$\forall n ~~ \sum_{k=0}^n 2^{-k} \le 2 - 2^{-n}$$
Then induction may be used directly:
$$\sum_{k=0}^{n+1} 2^{-k} \le 2 - 2^{-n - 1}$$
$$\sum_{k=0}^n 2^{-k} + 2^{-n-1} \le 2 - 2^{-n - 1}$$
$$\sum_{k=0}^n 2^{-k}\le 2 - 2^{-n}$$
Obviously you could see that it is a geometric series, but that is a generalization also.  

problems that become easier when you formulate them in a more general (or ambitious) form

The potential difficulty of a generalization isn't the only disadvantage.  If you disprove a generalization, then you haven't disproven the original theorem.  In that respect, a generalization effectively forces you to pick sides in the investigation of a theorem.